123 lines
3.2 KiB
TeX
123 lines
3.2 KiB
TeX
\section{Definable Sets}
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Armed with $(), \land, \lor, \lnot, \rightarrow, \forall,$ and $\exists$, we have enough tools to define sets.
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\definition{Set-Builder Notation}
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Say we have a condition $c$. \par
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The set of all elements that satisfy that condition can be written as follows:
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$$
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\{ x ~|~ \text{$c$ is true} \}
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$$
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This is read \say{The set of $x$ where $c$ is true} or \say{The set of $x$ that satisfy $c$.}
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\vspace{2mm}
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For example, take the formula $\varphi(x) = \exists y ~ (y + y = x)$. \par
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The set of all even integers can then be written
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$$
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\{ x ~|~ \varphi(x) \} = \{ x ~|~ \exists y ~ (y + y = x) \}
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$$
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\definition{Definable Sets}
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Let $S$ be a structure with a universe $U$. \par
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We say a subset $M$ of $U$ is \textit{definable} if we can write a formula that is true for some $x$ iff $x \in M$.
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\vspace{4mm}
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For example, consider the structure $\Bigl( \mathbb{Z} ~\big|~ \{+\} \Bigr)$ \par
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\vspace{2mm}
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Only even numbers satisfy the formula $\varphi(x) = \exists y ~ (y + y = x)$, \par
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So we can define \say{the set of even numbers} as $\{ x ~|~ \exists y ~ (y + y = x) \}$. \par
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Remember---we can only use symbols that are available in our structure!
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\problem{}
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Define the set of prime numbers in $\Bigl( \mathbb{Z} ~\big|~ \{\times, \div, <\} \Bigr)$
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\vfill
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\problem{}
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Define $\{0\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
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\vfill
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\problem{}
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Define $\{1\}$ in $\Bigl( \mathbb{Z}^+_0 ~\big|~ \{>\} \Bigr)$
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\vfill
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\pagebreak
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\problem{}
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Define the set of nonreal numbers in $\Bigl( \mathbb{C} ~\big|~ \{\text{real}(z)\} \Bigr)$ \par
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\hint{$\text{real}(z)$ gives the real part of a complex number: $\text{real}(3 + 2i) = 3$}
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\hint{$z$ is nonreal if $x \in \mathbb{C}$ and $x \notin \mathbb{R}$}
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\begin{solution}
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$\Bigl\{ x ~\bigl|~ \text{real}(x) = x \Bigr\}$
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\end{solution}
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\vfill
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\problem{}
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Define $\mathbb{R}^+_0$ in $\Bigl( \mathbb{R} ~\big|~ \{\times\} \Bigr)$ \par
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\vfill
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\problem{}
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Let $\bigtriangleup$ be a relational symbol. $a \bigtriangleup b$ holds iff $a$ divides $b$. \par
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Define the set of prime numbers in $\Bigl( \mathbb{Z}^+ ~\big|~ \{ \bigtriangleup \} \Bigr)$ \par
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\vfill
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\theorem{Lagrange's Four Square Theorem}
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Every natural number may be written as a sum of four integer squares.
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\problem{}
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Define $\mathbb{Z}^+_0$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
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\vfill
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\problem{}
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Define $<$ in $\Bigl( \mathbb{Z} ~\big|~ \{\times, +\} \Bigr)$
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\vfill
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\pagebreak
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\problem{}
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Consider the structure $S = ( \mathbb{R} ~|~ \{0, \diamond \} )$ \par
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The relation $a \diamond b$ holds if $| a - b | = 1$
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\problempart{}
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Define $\{\}$ in $S$.
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\problempart{}
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Define ${-1, 1}$ in $S$.
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\problempart{}
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Define $\{-2, 2\}$ in $S$.
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\vfill
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\problem{}
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Let $P$ be the set of all subsets of $\mathbb{Z}^+_0$. This is called a \textit{power set}. \par
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Let $S$ be the stucture $( P ~|~ \{\subseteq\} )$ \par
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\problempart{}
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Show that the empty set is definable in $S$.
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\problempart{}
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Let $x \Bumpeq y$ be a relation on $P$. $x \Bumpeq y$ holds if $x \cap y \neq \{\}$. \par
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Show that $\Bumpeq$ is definable in $S$.
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\problempart{}
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Let $f$ be a function on $P$ defined by $f(x) = \mathbb{Z}^+_0 - x$. This is called the \textit{compliment} of the set $x$. \par
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Show that $f$ is definable in $S$.
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\vfill
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\pagebreak |